Density Forecasts in Panel Models: A semiparametric Bayesian Perspective

This paper constructs individual-specific density forecasts for a panel of firms or households using a dynamic linear model with common and heterogeneous coeficients and cross-sectional heteroskedasticity. The panel considered in this paper features large cross-sectional dimension (N) but short time series (T). Due to short T, traditional methods have difficulty in disentanglingthe heterogeneous parameters from the shocks, which contaminates the estimates of the heterogeneous parameters. To tackle this problem, I assume that there is an underlying distribution of heterogeneous parameters, model this distribution nonparametrically allowing for correlation between heterogeneous parameters and initial conditions as well as individual-specific regressors, and then estimate this distribution by pooling the information from the whole cross-section together. I develop a simulation-based posterior sampling algorithm specifically addressing the nonparametric density estimation of unobserved heterogeneous parameters. I prove that both the estimated common parameters and the estimated distribution of the heterogeneous parameters achieve posterior consistency, and that the density forecasts asymptotically converge to the oracle forecast, an (infeasible) benchmark that is defined as the individual-specific posterior predictive distribution under the assumption that the common parameters and the distribution of the heterogeneous parameters are known. Monte Carlo simulations demonstrate improvements in density forecasts relative to alternative approaches. An application to young firm dynamics also shows that the proposed predictor provides more accurate density predictions.