M(παμγ)

We present proof of the inconsistency of the QMLE defined by Cho and White (2007). Inconsistency arises from the fact that the quasi-log-likelihood does not attain a maximum at the population parameter values. S1. PROOF OF INCONSISTENCY OF A QMLE IN THIS SUPPLEMENTAL section we prove that, for an autoregressive process, the gradient of the quasi-log-likelihood does not equal zero when evaluated at the population parameter values. Recall the mixture model for an autoregres-sive process (1 −π) ·N(θ1 + αXt−11)+πN(θ2 + αXt−11)(S1) on which the quasi-likelihood is based. For θ1 = μ and θ2 = μ+ γ, we have 1 n n∑ t=1 E[lt(παμγ)] = E[lt(παμγ)]:=M(παμγ) where M(παμγ) = E log π exp γ(Xt − αXt−1 −μ) −