Information ratio test for model misspecification on parametric structures in stochastic diffusion models

This document is prepared to provide the detailed and complete proof of the lemma in section 2.2. The proof of the lemma is sketched in the appendix due to the space limitations. The main theorem follows immediately from the lemma. Proposition 1. If conditions (a) and (b) in the appendix are satisfied, thenˆS = 1 n n ∑ k=1 g v (X (k−1)∆ , X k∆ ; θ, γ). By conditions (a) and (b), we havê θ pr → θ 0 and˜γ pr → γ 0. Applying the uniform law of large number (Theorem 4.1 in Wooldridge (1994)), the conclusions of proposition 1 are proved. Furthermore, note that under the null hypothesis of correct model specification, S(θ 0 , γ 0) = V(θ 0 , γ 0). Thus, by condition (d) and Slutsky's theorem, we have tr S −1 (ˆ θ, ˜ γ)V(ˆ θ, ˜ γ) pr → p, as n → ∞. Now, we provide a detailed and complete proof the lemma.