Transformations of the State Variable and Learning Dynamics
This article studies dynamics in a model where agents forecast a one dimensional state variable via ordinary least squares regressions on the lagged values of the state variable. We study the stability properties of alternative transformations of the state variable that the agent can endogenously set forth. We study the consequences on the economy's stability of the typical transformations that an econometrician would attempt, such as differencing, detrending, or taking instantaneous concave transformations, such as logarithms. Surprisingly, for the considered class of economies, we found that these transformations are destabilizing, whereas alternative transformations, which an econometrician would never consider, such as convex transformations, are stabilizing. Therefore, we ironically find that in our set-up, an active agent, who is concerned about learning the economy's dynamics and, in an attempt to improve forecasting, transforms the state variable using the standard transformations, is more likely to deviate from the steady state than a passive agent.
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