Reexamining an old story: uncovering the hidden small sample bias in AR(1) models

The first order autoregressive [AR(1)] model is widely used to investigate psycholog-ical dynamics. This study focuses on the estimation and inference of the autoregressive(AR) effect in AR(1) models under a limited sample size—a common scenario in psy-chological research. State-of-the-art estimators of the autoregressive effect are knownto be biased when sample sizes are small. We analytically demonstrate the causesand consequences of this small sample bias on the estimation of the AR effect, itsvariance, and the AR(1) model’s intercept, particularly when using OLS. In addition,we reviewed various bias correction methods proposed in the time series literature. Asimulation study compares the OLS estimator with these correction methods in termsof estimation accuracy and inference. The main result indicates that the small sam-ple bias of the OLS estimator of the autoregressive effect is a consequence of limitedinformation and correcting for this bias without more information always induces abias-variance trade-off. Nevertheless, correction methods discussed in this researchmay offer improved statistical power under moderate sample sizes when the primaryresearch goal is hypothesis testing.