Testing for One-Factor Models versus Stochastic Volatility Models

process ft (resp. X_t) around point a, over the period [0,1]. Thus,
_ L(n-1)r^j

--X~ Σ (^g ( ^fi∕ⁿ ) - ^g (^Xi/n ))
ⁿ
г =1

diverges (to either -∞ or to ∞), at rate √m, provided that L_χ(r, a) — Lf (r, a) = 0 (almost
surely) for all a ∈ A, with A having non-zero Lebesgue measure, that is provided that f_t and
X_t have different occupation densities over a non-negligible set.

Finally, U_n,_m,r can be written as

_ I (n-1)rj / 1 V^n-11                   _n (r⁽j^+1)/n ( ( f ) — ∩( X ))d _sʌ

1            n^ξn <^xj=¹ {∣Xj∕n^-Xi∕n∣<^ξn} ∖Jn∕n     ∖^g ∖Js) 9∖ s)) J

ⁿ “ I                  Lx ^{(1 ,X}i/n^{) + o}P⁽¹⁾

Expanding the sums, (28) can be rewritten as

1  /ⁿ (ʃɪ/n⁽ g^{( fs} )^- g^{( Xs} ))^{d s})

^nξn I     Lχ ^{(1 ,X} 1 /n^{) + o}P ⁽¹⁾
₊¹£IX/-X/|<jn}nj71/⁽g^(fS)g^(Xs))ds)

LX ^{(1 ,X} 2/n ) ^{+ o}P ⁽¹⁾

1{|x/-X[(n-i)_r/n]|<n}nj71/n⁽g^(fs^g^(Xs))ds)

LX ^{(1, x}(n-1)/n) ^{+ o}P ⁽¹⁾

_l 1 /¹ {∣χ_j∕n-χι∕n∣<ξn}ⁿ (j∕χ+1 // ^{(g (fs})   ^{g (Xs))ds})

^nξn I               LX^{(1 ,x} 1 /n^{) + o}P⁽¹⁾

_ι ¹ {∣χ_j∕_n-χ2∕n∣<ξn}ⁿ (ʃj/n+¹)^{/n (g (fs})   ^{g (Xs}))^ds)

LX ^{(1 ,X} 2 /n ) ^{+ o}P ⁽¹⁾

1r∣_{r x} ∖_f∖n (f^{(j+1) /n} ( g ( f_s ) —g ( X_s ))d s ^∖
∣       {∣^χj∕n^-χ[(n-1)r∕n] ∣<^ξn} \/n/^{n      si}''¹' ^{sγ s}

LX ^{(1, x}(n-1)/n) ^{+ o}P ⁽¹⁾

+...

+ ɪ / ' χ χ . ⁿ '■          .^{g fS}     .^g Xs ^ds

^nξn I                LX ^{(1 ,X} 1 /n^{)+ o}P ⁽¹⁾

. X n ʃ ■ 's g - -

LX ^{(1 ,x}2/n ) ^{+ o}P ⁽¹⁾

¹                   ^ʃ         '    ■ '   ■' ∖

LX ^{(1 ,x} ( n-1) /n ) ^{+ o}P ⁽¹⁾

26

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