Testing for One-Factor Models versus Stochastic Volatility Models

_r- L(n-1)rJ / ^y^n-¹1 _rι

+ √m ^v^j=¹ {∣X∙∕∙

n ⅛ I

i =1 \

-X√_n∣<ξn} ²ⁿ ʃ/n') / (^X^s^Xj/n ) ^μ⁽^X^s^)d^S

y^n-¹ 1

X₃=1 ¹ {∣Xj∕n-Xi∕n∣<ξn}

S n,m,r

ч.

L ( n-1) rJ ∕^yⁿ-¹ 1

= ¹ {l^χj∕n^-χi∕n∣<^ξn

∑n-1
j =1

}ⁿ (ʃj/r+¹)^/n⁽ g⁽^fs )^- g⁽^fi∕n⁾⁾^d^s) ʌ

^{} v J/}. (25)

¹ {IX∙∕n-X_i∕n∣<ξn}

Qn,m,r

Now, P_n,_m,_r is O_p(1), again for the same argument used in the proof of part (i)a Step 1, and
similarly S_n,_m,_r = o_P (1), since it has the same behavior under both Ho and H a . Then, we
need to show that Q_n,_m,_r, in absolute value, diverges. Now,

L ( n-¹) rJ ^yⁿ— 11

¹ _Z) _ ¹ v ³=¹ υX∙∕n

√.∙ = n I

i=1

L ( n- 1) rJ y^n-¹1

₊1 ^v j=¹^{^χ

n I

i =1 \

ч_______________________________________________

L (n-1) rJ ( y

¹^vn

i=1

n-1

= = ¹ ^^χj∕n^-

-χ,∕,∣<tn}n (C^υ^/n (g (^χs) - g (^Xi/n))^ds)

^y n-1

=1 ¹ {∣Xj∕n-Xi∕n∣<ξn}

T n,m,r

-Xi,,∖<ξn}n (C^j^/n (g ( fs) - g (Xs))^ds)

^y n-1

=1 ¹ {∣X∙∕n-Xi∕n∣<ξn}

^U n,m,r

∙X_i∕n∣<ξn} (^g (^fi/n) ^{- g} (^Xi/n))

∑n-1

j = 1 ¹ {∣Xj∕n-Xi∕n∣<ξn}

V n,m,r

(26)

Note that Т_п_тг is o_P(1), by the same argument as the one used in part (i)a, Step 1.

As for V_n,_m,_r, it can be rewritten as

₁ L(^n-¹) ^rJ / r r ʌ

^ ∑ (g (^fi∕n) - g (^χi∕n)) = Çyθ g⁽^χs^)ds^- yɔ g⁽^fs^)dSJ ⁺ OP⁽ⁿ-¹ /²⁾^,

(27)

where the first term of the right hand side of (27) is almost surely different from 0, given that
X_s and f_s have different occupation density. Also, in the case in which f_s is one-dimensional,
we have that

1 ^l⁽ⁿ 1)^rJ ∞ ∞

- ∑ (g (fi/n) - g (Xi/n)) = g(a)Lx(r,a)da - g(a)Lf (r,a)da + Op(n^-¹^/²),

ⁿ /_i J-OQ -∞

where Lf (r,a) (resp. L_x(r,a)) denotes the standardized local time of the process f_t (resp.
X_t) evaluated at time r and at point a, that is it denotes the amount of time spent by the

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Testing for One-Factor Models versus Stochastic Volatility Models

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