# SFB 303 Discussion Paper No. B - 225

**Author**: Zenner, Markus

**Title**: Performance of Least Squares Learning in Autoregressive Models with Forecast Feedback. The Stochastic
Case

**Abstract**: In a recent paper we considered in a deterministic framework a simple autoregressive model with forecast
feedback together with a least squares learning procedure in order to find out under what conditions there is
convergence of the learning procedure towards rational expectations. Here we consider the same model in a
stochastic framework by introducing a disturbance term into the model equation. Although the concept of rational
expectations makes only little sense in a deterministic framework, because it is identical with perfect foresight, it
is useful to analyze the deterministic least squares algorithm (with means due only to the deterministic case)
because there is a close connection between the behaviour of the deterministic and the stochastic algorithm. In
this paper we analyze this connection and determine the asymptotic behaviour of the stochastic algorithm.
Because of the lack of theoretical results due to the unpleasant structure of the algorithm, parts of this paper are
only descriptive using the results of various computer simulations. Nevertheless we can give heuristical
explanations for the observed behaviour of the least squares algorithm. This paper is organized as follows:
Section 2 gives the model specification, determines the rational expectations and gives necessary and sufficient
conditions for existence. In Section 3 we prove that if the learning procedure is convergent then the resulting
limit expectations are rational. Section 4 is concerned with the long-term behaviour of the stochastic algorithm,
reviews shortly the results for the deterministic algorithm and analyses the connection between the two
algorithms. Section 5 provides a summary of our findings and some concluding remarks.

**Keywords**:

**JEL-Classification-Number**:

**Creation-Date**: November 1992

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