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 ))
ⁿг =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

(28)

ⁿ “ I Lx ⁽¹^,Xi/n^{) +}^oP⁽¹⁾

Expanding the sums, (28) can be rewritten as

1 /ⁿ (ʃɪ/n⁽ g⁽^f^s )^- g⁽^X^s ))^d^s)

^nξn I Lχ ⁽¹^,X 1 /n^{) +}^oP ⁽¹⁾
₊¹£IX/-X/|<jn}nj71/⁽g⁽^f^S⁾g⁽^X^s^))ds)

LX ⁽¹^,X 2/n ) ⁺^oP ⁽¹⁾

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

LX ⁽¹^{, x}(n-1)/n) ⁺^oP ⁽¹⁾

_l 1 /¹ {∣χ_j∕n-χι∕n∣<ξn}ⁿ (j∕χ+1 // ⁽^g⁽^f^s) ^g⁽^X^s^))d^s)

^nξn I LX⁽¹^,x 1 /n^{) +}^oP⁽¹⁾

_ι ¹ {∣χ_j∕_n-χ2∕n∣<ξn}ⁿ (ʃj/n+¹)^/n⁽^g⁽^f^s) ^g⁽^X^s))^d^s)

LX ⁽¹^,X 2 /n ) ⁺^oP ⁽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) ⁺^oP ⁽¹⁾

+...

+ ɪ / ' χ χ . ⁿ '■ .^{g f}^S .^g Xs ^d^s

^nξn I LX ⁽¹^,X 1 /n⁾⁺^oP ⁽¹⁾

. X n ʃ ■ 's g - -

LX ⁽¹^,x2/n ) ⁺^oP ⁽¹⁾

¹ ^ʃ ' ■ ' ■' ∖

LX ⁽¹^,x ( n-1) /n ) ⁺^oP ⁽¹⁾

CorPapers

Testing for One-Factor Models versus Stochastic Volatility Models

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