Path dependence in models with fading memory or adaptive learning

We consider a learning mechanism where expected values of an economic variable in discrete time are computed in the form of a weighted average that exponentially discounts older data. Also adaptive expectations can be expressed as weighted sums of infinitely many past states, with exponentially decreasing weights, but these are not averages since the weights do not sum up to one for any given initial time. These two different kinds of learning, which are often considered as equivalent in the literature, are compared in this paper. The statistical learning dynamics with exponentially decreasing weights can be reduced to the study of a two-dimensional autonomous dynamical system, whose limiting sets are the same as those obtained with adaptive expectations. However, starting from a given initial condition, different transient dynamics are obtained, and consequently convergence to different attracting sets may occur. In other words, even if the two different kinds of learning dynamics have the same attracting sets, they may have different basins of attraction. This implies that local stability results are not sufficient to select the kind of long-run dynamics since this may crucially depend on the initial conditions.We show that the two-dimensional discrete dynamical system equivalent to the statistical learning with fading memory is represented by a triangular map with denominator which vanishes along a line, and this gives rise to particular structures of their basins of attraction, whose study requires a global analysis of the map.We discuss some examples motivated by the economic literature.