The purpose of this paper is twofold: In the first part, it derives a new set of sufficient and necessary conditions for strongly consistent ordinary least-squares (OLS) estimation in the simple linear regression model with stochastic regressors. Our primary condition is weaker than what is presently available in the literature. In return, the error terms are assumed to form an independent Gaussian sequence and the stochastic regressors are required to meet a particular measurability condition. In the second part of the paper, we turn to OLS estimation of the structural parameters in an economic model in which the premise that agents are rational is replaced by the assumption that they form expectations by means of adaptive learning. Research in this area is still in its infancy, with most of the learning literature having so far been concerned with the convergence behaviour of the learning routine as such. In particular, we consider learning to follow a decreasing-gain algorithm, resulting in the asymptotic collinearity of the regressors.
While existing results in the literature on strong consistency are inapplicable to this model, we appeal to our new consistency proof to show that OLS can indeed asymptotically identify the structural parameters.