SFB 303 Discussion Paper No. B - 224

Author: Zenner, Markus
Title: Performance of Least Squares Learning in Autoregressive Models with Forecast Feedback. The Deterministic Case
Abstract: In this paper we try to analyse such a simple model where the current endogenous variable is determined as a linear combination of the one-period lagged endogenous variable and the forecast of a future value of the endogenous variable. This model was already considered by KOTTMANN, KULIBERDA (1990) in a Monte- Carlo-study. We used their simulation computer program to get a deeper understanding of the quite complicated dynamics of that model and discovered some interesting facts beyond the study of KOTTMANN, KULIBERDA: The simple model is able to produce not only convergence and divergence but also stable limit cycles and chaos, a fact which has not yet been observed for linear models with OLS-learning. Because of the complicated dynamics we restrict ourselves in this paper to the deterministic case. Thus no random shocks or noise appear in our model. Nevertheless, the remaining dynamics are complicated enough to produce immense difficulties in determining or predicting the behaviour of a path starting with known initial values. Therefore, parts of this paper are only descriptive, but we have tried to prove conclusive results whenever they could be achieved with a moderate amount of technical details. The paper is organized as follows: Section 2 gives the model specification, determines the rational expectations and gives necessary and sufficient conditions for their existence. In Section 3 the learning algorithm is transformed into a system of nonlinear difference equations and existence and stability of equilibria of that system are determined. Section 4 describes the long-term behaviour of the learning algorithm based on the results of various simulations. Some conclusive results are proved. Section 5 provides a discussion of generalisations and some concluding remarks.
Creation-Date: November 1992
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