Testing for One-Factor Models versus Stochastic Volatility Models

(b) Let S →∞. Suppose that, as n, m, ξ_n^-1 →∞, nξn →∞ and nξ_n² → 0 , and, for any ε > 0
arbitrarily small, m/n^{1 -ε} → 0; then do not reject H₀ if

Zn,m ≤ CVα^S

and reject otherwise. This rule provides a test with asymptotic size equal to α and asymptotic
unit power.

As mentioned above, our test is designed to compare two classes of models, namely the one-
factor diffusion models and the stochastic volatility models, regardless of the specification of the
drift term. Therefore, if for example model (1) is augmented by adding another factor into the drift
term (see e.g. Hull and White, 1994), our test will still fail to reject the null hypothesis considered,
because the drift term is, over a fixed time span, of a smaller order of probability than the diffusion
term and so is asymptotically negligible.

2.3 Market Microstructures and jumps

The asymptotic theory derived in the previous subsection relies on the fact that the underlying
process is a continuous semi-martingale. However, some recent financial literature has pointed
out the effects of possible jumps and market microstructure error on realized volatility (see e.g.
Barndorff-Nielsen and Shephard, 2004c,d, Corradi and Distaso, 2004, Andersen, Bollerslev and
Diebold, 2003 for jumps, and Alt-Sahalia, Mykland and Zhang, 2003, Zhang, Mykland and Alt-
Sahalia, 2003, Bandi and Russell, 2003, Hansen and Lunde, 2004 for microstructure noise).

We begin by analyzing the contribution of large and rare jumps. Suppose that the generating
process in (1) is augmented by a jump component,

d Xt = μ ( Xt )d t + d zt + σt d W1 ,t,

where σt = σ(Xt), and zt is a pure jump process.

The test statistics based on Zn,m,r are not robust to the presence of jumps. The intuitive reason
is that jumps have a different impact on the two components of the statistics, namely

L (^n-1) ^rJ

n^-1        S_n² (X_i/_n) and RVm,r .

i=1

In fact, in the presence of jumps, RVm,r converges to the integrated volatility process plus the sum
of the squared magnitudes of the jumps (see Barndorff-Nielsen and Shephard, 2004c). Conversely,
n^- 1 Σ!'ni—1)^rJ Sn(Xi/n) converges to integrated volatility plus the weighted sum of the squared
magnitudes of the jumps, where the weights depend on the local time of Xt . Broadly speaking, a

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