We establish consistency and asymptotic normality of the quasi-maximum likelihood estimator in the linear ARCH model. Contrary to the existing literature, we allow the parameters to be in the region where no stationary version of the process exists. This implies that the estimator is always asymptotically normal.

This paper discusses a consistent bootstrap implementation of the likelihood ratio (LR) co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model that obtain under the reduced rank null hypothesis. A full asymptotic theory is provided that shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co-integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice.

THIS SUPPLEMENT CONTAINS proofs of the theoretical results stated in Cavaliere, In addition, extended details and discussions (also covering models with intercept) are given for the Monte Carlo results reported in Section 4 of

Based on the GARCH literature, Engle and Russell (1998) established consistency and asymptotic normality of the QMLE for the autoregressive conditional duration (ACD) model, assuming strict stationarity and ergodicity of the durations. Using novel arguments based on renewal process theory, we show that their results hold under the stronger requirement that durations have finite expectation. However, we demonstrate that this is not always the case under the assumption of stationary and ergodic durations. Specifically, we provide a counterexample where the MLE is asymptotically mixed normal and converges at a rate significantly slower than usual. The main difference between ACD and GARCH asymptotics is that the former must account for the number of durations in a given time span being random. As a by‐product, we present a new lemma which can be applied to analyze asymptotic properties of extremum estimators when the number of observations is random.