On plug-in estimation of long memory models

We consider the Gaussian ARFIMA(j, d, l) model, with spectral density f(theta)(lambda), theta is an element of Theta subset of R-p, lambda is an element of (-pi,pi), d is an element of (0,(1)/(2)) and an unknown mean mu is an element of R. For this class of models, the n(-1)-normalized information matrix of the full parameter vector, (mu,theta), is asymptotically degenerate. To estimate theta, Dahlhaus (1989, Annals of Statistics 17, 1749-1766) suggested using the maximizer of the plug-in log-likelihood, L-n(theta,(mu) over tilde (n)), where (mu) over tilde (n), is any n((1-2d)/2)-consistent estimator of mu. The resulting estimator is a plug-in maximum likelihood estimator (PMLE). This estimator is asymptotically normal, efficient, and consistent, but in finite samples it has some serious drawbacks. Primarily, none of the Bartlett identities associated with L-n(theta,(mu) over tilde (n)) are satisfied for fixed n. Cheung and Diebold (1994, Journal of Econometrics 62, 301-316) conducted a Monte Carlo simulation study and reported that the bias of the PMLE is about 3-4 times the bias of the regular maximum likelihood estimator (MLE). In this paper, we derive asymptotic expansions for the PMLE and show that its second-order bias is contaminated by an additional term, which does not exist in regular cases. This term arises because of the failure of the first Bartlett identity to hold and seems to explain Cheung and Diebold's simulated results. We derive similar expansions for the Whittle MLE, which is another estimator tacitly using the plug-in principle. An application to the ARFIMA(0,d,0) shows that the additional bias terms are considerable.