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Editor's Special Invited Paper: Sequential Estimation for Time Series Models

Abstract 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 the regret in using a sequential estimator of β can be significantly negative for 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.