Testing for One-Factor Models versus Stochastic Volatility Models

2.2 Limiting Behavior of the Statistic

We can now establish the limiting distribution of the proposed test statistics based on Zn,m,r ,
defined in (3), for both the cases where n = m and m/n → 0, as m, n →∞.

Theorem 1. Let Assumption 1 hold.

Under H0,

(i)a if, as n,m, ξ-¹ → ∞, nξ_n → ∞ and for any arbitrarily small ε > 0, n¹ /2+^εξ_n → 0, and if
m = n, then, pointwise in r ∈ (0, 1)

Zn,r -→ Zr - MN (θ, 2 [∞ σ⁴(a) LX⁽r^, a⁾⁽LX(ɪ^,a)^- LX⁽r^, a)) da} ,        (6)

_∞                LX (1, a)

where Z_n,r ≡ Z_n,n,r and

ɪɪ ^r         ₂

LX^(r,a) = ψim ψ _σ2(a) yθ ¹ {Xu∈[α,α+ψ]}^{σ (}Xu)^du

denotes the standardized local time of the process Xt .

(i)b Define Z_n = maxj=₁ ,...,J ∣ Z_n,_rj ∣ and Z = max_j∙=₁ ,...,J ∣ Zr_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 →∞,
nξ_n → ∞, and, for any ε > 0 arbitrarily small, n¹ /2+^εξ_n → 0, and if m = n, then

Z_n -→ Z,

where ∀ r,r',

LX (min( r, r' ) ,a )( LX (1 ,a ) - LX (min( r, r' ) ,a ))d

LX (1 ,a )                     a.

(i)c If, as n, m, ξ_n ¹ → ∞, nξ_n → ∞ and nξn → 0, and, for any ε > 0 arbitrarily small, m/n^{1 ε} →

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

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