Testing for One-Factor Models versus Stochastic Volatility Models

(i)c The statistic Z_n,_m,_r can be rewritten as

√— L (ⁿ-¹) ^rJ

—   £  (S²_n(Х_г/_п) - σ²(Х_г/_п))

i =1

^s--------------------------------S/--------------------------------'

An,m,r

—

L ( m— 1) rJ

Σ

j=1

'm ^{- X}j/m) ^{- σ σ2(X}s^)dS
JO

S/

Bm,r

√— ^{l ( n-1})^rJ                    r

+— ∑ σ²(Xi/n) - V— I σ²(Xs)ds.

ⁿ i=1                    ^0o

'-----------------------------------_v-----------------------------------'

C n,m,r

(23)

Note that A_n,_m,_r = op(1) by Lemma 2.

We first need to show that C_n,_m,_r = o_a._s.(1). Given Assumption 1(a), Lemma 1, and recalling
the modulus of continuity of a diffusion (see McKean, 1969, pp.95-96),

—— ^{l (} n- i^)rJ

— 52 σ²(Xi/n) -V—   σ²(Xs)ds

ⁿ TX                   -ʃo

^l— ^{l ( n-1})^rJ

— Σ σ ²( Xi/n )
i=1

^{l ( n-1})^rJ _r( i+1) /n

— ∑        σ² ( X_s )d s

i=1    ^i/n

(σ²(Xi/n) - σ²(Xs)) ds

^{l ( n-1})^rJ _r( i+1) /n

— ∑

i=1 ^i/n

^{L ( n-1})^rJ _r( i +1) /n

- σ²(Xs)∣ ds

V— sup   ∣σ²(X_s) - σ²(X_τ)∣ ≤ V— sup ∣Vσ²(X_τ)∣   sup  ∣X_s - X_τ∣

|s—τ ∣≤ 1 /n                                        τ ∈ [0 ,r ]                 |s—τ ∣≤ 1 /n

s∈ [θ ,r ]                                                                                   s∈ [θ ,r ]

V—iC>₍₁._S( (n^ε/²)Oa.s. (n ¹ /² log n) = 0a.s. (1),

as n¹ /^{2 ε}/²n ¹ /² log n → 0. Thus,

Zn,m,r

Bm,r + ⁰a.s.⁽¹⁾.