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Suppose Z t is the square of a time series Y t whose conditional mean is zero. We do not specify a model for Y t, but assume that there exists a p×1 parameter vector Φ such that the conditional distribution of Z t| Z t−1 is the same as that of [ABSTRACT FROM AUTHOR]

This article contains our responses to the eight discussion pieces written by experts, commenting on the results presented in Sriram and Iaci (2014). The majority of the discussants described complementary sequential problems arising in time series and other models involving dependent data, while two of the discussants provided thoughtful views on the phenomenon ofnegative regretinvestigated in our article. Prompted by an excellent observation made in one of the discussion pieces, we have attempted to shed further light on the phenomenon ofnegative regretin the context of sequential estimation of the autoregressive parameter. [ABSTRACT FROM AUTHOR]

This article revisits sequential estimation of the autoregressive parameter β in a first-order autoregressive (AR(1)) model and construction of a sequential confidence region for a parameter vector θ in a first-order threshold autoregressive (TAR(1)) model. To resolve a theoretical conjecture raised in Sriram (1986), we provide a comprehensive numerical study that strongly suggests that theregretin using a sequential estimator of β can be significantlynegativefor many heavy-tailed error distributions and even for normal errors. Secondly, to investigate yet another conjecture about the limiting distribution of a sequential pivotal quantity for θ in a TAR(1) model, we conduct an extensive numerical study that strongly suggests that the sequential confidence region has much better coverage probability than that of a fixed sample counterpart, regardless of whether the θ values are inside or on or near the boundary of the ergodic region of the series. These highlight the usefulness of sequential sampling methods in fitting linear and nonlinear time series models. [ABSTRACT FROM AUTHOR]