Estimation of constant gain learning models

Constant Gain Learning models present a unique set of challenges to the econometrician compared to models based on Rational Expectations. This is because of the differences in the equilibrium concept and stability conditions between the two paradigms. This paper focuses on four key issues:

(i)     Stability conditions – The stability properties of these models play a vital role in the estimation process, especially in the case of lagged endogenous variables when a multiplicity of solutions exist

(ii)    Identification – With small values of the (constant) gain parameter, structural parameters that are not well identified under rational expectations may pose a similar problem under learning

(iii)   Likelihood function – Because the learning parameters (that are functions of the unobserved shocks) are entangled nonlinearly in the sate space model, deriving the joint density of the data as a function of only the model parameters (but marginalized of the learning parameters) is no longer feasible with the aid of the Kalman filter

(iv)   Initialization of the learning parameters – Can be accomplished by estimating them based on either a training sample or the full sample

This paper provides an in-depth discussion of these issues and provides an accessible estimation guide that is helpful to the applied researcher. The estimation methodology is complemented by three illustrative examples. The first two emphasize these points with the aid of simulated data whereas the third deals with a small New Keynesian model that is taken to real data. For ease of replicability of the results, great care is taken to present complete details of the estimation procedure. As demonstrated by these examples, the researcher is free to use either approach for estimating the initial value of the learning parameters so long as the data carries enough information about the structural parameters of the model.