Testing for One-Factor Models versus Stochastic Volatility Models

where σ_s⁴ = σ⁴(Xs) under H0 and σ_s⁴ = g²(fs) under HA; in other words the estimator defined in
(11) is consistent for the “true” integrated quarticity under both hypotheses and therefore provides
an estimator of the upper bound of the term in (9).

On the other hand, we shall provide correct asymptotic critical values for the limiting distribu-
tion in Theorem 1, parts (i)c and (i)d and in Corollary 1. In order to obtain asymptotically valid
critical values and to make the limit theory derived in Theorem 1 part (i)d feasible, we will use a
data-dependent approach. For s =1, . . . , S, where S denotes the number of replications, let

2 L^(m-1)^rjJ

3 Σ2 m (X(i+1) /m — Xi/m)

is a consistent estimator of twice the integrated quarticity and, for each s, ( η^{( s} ) η2 ^s )... ηJ^s ) ) is
drawn from a N(0, IJ). Then compute maxj=₁ ,...,J ∣dm,_r∣ , repeat this step S times, and construct
the empirical distribution. As S → ∞, the empirical distribution of max_j∙=₁ ,...,J ∣dm)r∣ will converge
the distribution of a random variable defined as

Therefore an asymptotically valid critical value for the limit theory in Theorem 1 part (i)d will be
given by CVaS, which denotes the (1—α) — quantile of the empirical distribution of max_j∙=₁ ,...,J ∣cm_l)j |,
computed using S replications. Given the discussion above, CV_α^S will provide an upper bound for
the critical values of the limiting distribution derived in Theorem 1, part i(b). The implied rules
for deciding between H0 and HA are outlined in the following Proposition.

Proposition 1. Let Assumption 1 hold.

(a) Let S →∞. Suppose that as n, m, ξ_n^-1 →∞, nξn →∞ and, for any ε > 0 arbitrarily small,
n¹ /2^+εξn → 0. If m = n, then do not reject H0 if

Zn ≤ CVα^S

and reject otherwise. This rule provides a test with asymptotic size smaller than α and
asymptotic unit power.

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