Corrected confidence intervals for parameters in adaptive linear models

Consider an adaptive linear model y t = x t ′ θ + σ e t , where x t = ( x t 1 , … , x tp ) ′ may depend on previous responses. Woodroofe and Coad [1999. Corrected confidence sets for sequentially designed experiments: examples. In: Ghosh, S. (Ed.), Multivariate Analysis, Design of Experiments, and Survey Sampling. Marcel Dekker, Inc., New York, pp. 135–161] derived very weak asymptotic expansions for the distributions of an appropriate pivotal quantity and constructed corrected confidence sets for θ , where the correction terms involve the limit of ∑ t = 1 n x t x t ′ / n (as n approaches infinity) and its derivatives with respect to θ . However, the analytic form of this limit and its derivatives may not be tractable in some models. This paper proposes a numerical method to approximate the correction terms. For the resulting approximate pivot, we show that under mild conditions the error induced by numerical approximation is o p ( 1 / n ) . Then, we assess the accuracy of the proposed method by an autoregressive model and a threshold autoregressive model.