Optimal trade-off between sample size, precision of supervision, and selection probabilities for the unbalanced fixed effects panel data model

This paper is focused on the unbalanced fixed effects panel data model. This is a linear regression model able to represent unobserved heterogeneity in the data, by allowing each two distinct observational units to have possibly different numbers of associated observations. We specifically address the case in which the model includes the additional possibility of controlling the conditional variance of the output given the input and the selection probabilities of the different units per unit time. This is achieved by varying the cost associated with the supervision of each training example. Assuming an upper bound on the expected total supervision cost and fixing the expected number of observed units for each instant, we analyze and optimize the trade-off between sample size, precision of supervision (the reciprocal of the conditional variance of the output) and selection probabilities. This is obtained by formulating and solving a suitable optimization problem. The formulation of such a problem is based on a large-sample upper bound on the generalization error associated with the estimates of the parameters of the unbalanced fixed effects panel data model, conditioned on the training input dataset. We prove that, under appropriate assumptions, in some cases “many but bad” examples provide a smaller large-sample upper bound on the conditional generalization error than “few but good” ones, whereas in other cases the opposite occurs. We conclude discussing possible applications of the presented results, and extensions of the proposed optimization framework to other panel data models.