Testing for One-Factor Models versus Stochastic Volatility Models

-X√_n∣<ξn} ^{2 n} ʃ/n') / (^{Xs  X}j/n ) ^{μ ( Xs )d S}

y^n-1 1

X₃=1 ¹ {∣Xj∕n-Xi∕n∣<ξn}

S/

S n,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   ^{y n}— 11

¹ _Z) _  ¹ v      ³=¹ υX∙∕n

√.∙          = n I

i=1

ч

L ( n- 1) rJ   y^n-11

₊1 ^v     j=^{1 { χ}

n I

i =1   \

ч_______________________________________________

L (n-1) rJ ( y

^{1 v}
n

i=1

4

^y n-1

=1 ¹ {∣Xj∕n-Xi∕n∣<ξn}

V

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}

V

^U n,m,r

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

Note that Т_птг 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),

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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