Empirical likelihood confidence regions in time series models

Adjusted empirical likelihood for long-memory time-series models

“…Therefore, β − β n = O p (n − 1 2 ) which implies β = β n + n − 1 2 u with |u| < +∞. Since ψ j (I(ω j ), β n ) → ψ j (I(ω j ), β) in probability, from Monti (1997) we have, 1 n n j=1 ψ j ψ j = 1 n n j=1 ψ j (I(ω j ), β n )ψ j (I(ω j ), β n ) + O p (n − 1 2 ) =Σ(β n ) + O p (n − 1 2 ), (A.5) whereΣ (β n ) = 1 n n j=1 ψ j (I(ω j ), β n )ψ j (I(ω j ), β n ) .…”

“…3) Monti (1997) showed that this estimator has the interpretation of an M-estimator from asymptotically independent periodogram ordinates and applied the empirical likelihood to short-memory time series models. As pointed out by Yau (2012), the EL method used by Monti (1997) cannot be directly applied to long-memory scenario due to the dependence structure of the periodogram.…”

“…As pointed out by Yau (2012), the EL method used by Monti (1997) cannot be directly applied to long-memory scenario due to the dependence structure of the periodogram. As shown in Lemma 1 of Yau (2012), the periodogram ordinates are asymptotically independent which ensures that the empirical likelihood ratio statistic to be chi-squared distributed and uses the ψ j 's in equation (2.3) to construct the empirical likelihood.…”

“…Following Monti's (1997) argument, Yau (2012 extended the empirical likelihood ratio statistic to the ARFIMA(p,d,q) model defined by…”

“…Mykland (1995) established the connection between the dual likelihood and the empirical likelihood through the martingale estimating equations and applied it to time series model. Monti (1997) developed the idea of extending the EL method to short-memory stationary time series by using the Whittle's (1953) method to obtain an M-estimator of the periodogram ordinates of time series models which are asymptotically independent. However, his method can not be applied directly to long-time memory time series model.…”