Testing for One-Factor Models versus Stochastic Volatility Models

(i)d Define Z_n,_m = maxj=1 ,...,J ∣ Z_n,_m,r_j ∣ and ZM = max_j∙=1 ,...,J ∣ ZMr_j ∣, where 0 < r 1 < ... <

rj-1 <rj < . . . < rJ < 1, for j =1, . . . , J, with J arbitrarily large but finite. If, as

n,m, ξ_n 1 → ∞, nξn → ∞ and nξn → 0, and, for any ε > 0 arbitrarily small, m/n 1 ^ε → 0,

then

Z_n,m -→ ZM,

with

where that, ∀ r,r',

(ii) Under HA, if, as n, m, ξ_n^-1 →∞, nξn →∞and nξ_n² → 0, and if m/n → π ≥ 0, then,

where ς(ω) > 0 for all ω ∈ Ω⁺, where Ω⁺ is defined as in the statement of Ha.

Notice that, as shown in the proof in the Appendix, under the alternative hypothesis, and in
the case where ft is a one-dimensional process, the dominant term of the proposed statistic is
a scaled version of the absolute value of the difference between the local times of Xt and ft . If
instead ft is a multidimensional process, then the multivariate local time analogue of the Lf (1, a)
used in Theorem 1 is not defined, but it can still be interpreted as a occupation density of the
multivariate diffusion ft (see e.g. Geman and Horowitz, 1980 and Bandi and Moloche, 2001).
Therefore, in both cases, there exists an (almost surely) strictly positive random variable ς, such
that (1 /√m) ∣Z_m,_n,_r| ≥ ς, with probability approaching one.

The following Corollary considers the case where r = 1, i.e. when we use the whole span of
data in constructing the test statistic.

Corollary 1. Let Assumption 1 hold. Under H0, if, as n, m, ξ_n^-1 →∞, nξn →∞ and nξ_n² → 0,
and, for any ε > 0 arbitrarily small, m/n 1^-ε → 0, then

Zn,m,1 -^d→ MN 0,2 ^∞ σ⁴ (a) LX (1,a)da .

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