Strong consistency of the least squares estimator in regression models with adaptive learning

Norbert Christopeit, Michael Massmann

Research output: Contribution to JournalArticleAcademicpeer-review


This paper looks at the strong consistency of the ordinary least squares (OLS) estimator in linear regression models with adaptive learning. It is a companion to Christopeit & Massmann (2018) which considers the estimator’s convergence in distribution and its weak consistency in the same setting. Under constant gain learning, the model is closely related to stationary, (alternating) unit root or explosive autoregressive processes. Under decreasing gain learning, the regressors in the model are asymptotically collinear. The paper examines, first, the issue of strong convergence of the learning recursion: It is argued that, under constant gain learning, the recursion does not converge in any probabilistic sense, while for decreasing gain learning rates are derived at which the recursion converges almost surely to the rational expectations equilibrium. Secondly, the paper establishes the strong consistency of the OLS estimators, under both constant and decreasing gain learning, as well as rates at which the estimators converge almost surely. In the constant gain model, separate estimators for the intercept and slope parameters are juxtaposed to the joint estimator, drawing on the recent literature on explosive autoregressive models. Thirdly, it is emphasised that strong consistency is obtained in all models although the near-optimal condition for the strong consistency of OLS in linear regression models with stochastic regressors, established by Lai & Wei (1982a), is not always met.

Original languageEnglish
Pages (from-to)1646-1693
Number of pages48
JournalElectronic Journal of Statistics
Issue number1
Publication statusPublished - 1 Jan 2019


  • Adaptive learning
  • Almost sure convergence
  • Non-stationary regression
  • Ordinary least squares


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